Prolific Compositions

Abstract

Under what circumstances might every extension of a combinatorial structurecontain more copies of another one than the original did? This property, whichwe call prolificity, holds universally in some cases (e.g., finite linearorders) and only trivially in others (e.g., permutations). Integercompositions, or equivalently layered permutations, provide a middle ground. Inthat setting, there are prolific compositions for a given pattern if and onlyif that pattern begins and ends with 1. For each pattern, there is an easilyconstructed automaton that recognises prolific compositions for that pattern.Some instances where there is a unique minimal prolific composition for apattern are classified